Multiplication of polynomials with coefficients in Galois fields (GF) is widely used in communication systems for Reed Solomon (RS) coding and in advanced encryption standards (AES). In some, the basic Galois fields multiplication is not enough and a more advanced Galois fields operation like Galois fields multiply and accumulate (GF-MAC) or Galois fields multiply and add (GF_MPA) are needed. Galois field multiplication is difficult and time consuming for traditional digital signal processors (DSP) to perform. DSP's are optimized for finite impulse response (FIR) filtering and other multiply accumulate (MAC) intensive operations, but do not efficiently process Galois field types of operations. One approach uses a straight forward polynomial multiplication and division over the Galois field using linear feedback shift registers (LFSR's) which process one bit at a time. This is a very slow process. For example, in broadband communication for AES types of applications, where the bit rate is up to 40 megabits per second, there will be up to 5 million GF multiplications per second (GF-MPS) and each multiplication may require many e.g. 60–100 operations. Another approach uses look-up tables to perform the Galois field multiplication. Typically, this approach requires 10–20 or more cycles which for 5 GF-MPS results in a somewhat lower but still very large number of operations e.g. 20×5=100 MIPS or more. Reed-Solomon codes have been widely accepted as the preferred error control coding scheme for broadband networks. A programmable implementation of a Reed-Solomon encoder and decoder is an attractive solution as it offers the system designer the unique flexibility to trade-off the data bandwidth and the error correcting capability that is desired based on the condition of the channel. The first step in Reed-Solomon decoding is the computing of the syndromes. The syndromes can be formally defined as Si═R mod G where i=(0,1 . . . 15). The received code word may be expressed in polynomial form as Ri=roXN−1+r1XN−2+ . . . rN−1 where the length of the received word is N. It can be seen that computing the syndrome amounts to polynomial evaluation over Galois field at the roots as defined by the j'th power of the i'th root of the generator polynomial. For each received word in the Reed-Solomon Algorithm there are sixteen syndromes to be calculated which raise the operations by a factor of sixteen to 1.6 billion-operations per second (BOPS)-not practical on current microprocessors. Using the straight forward multiplication instead of the look-up tables raises the operation rate to 6.4 BOPS. The need for Galois field multiplications is increasing dramatically with the expansion of the communications field and the imposition of encryption requirements on the communication data. This further complicates the matter because each domain-error checking, encryption-needs Galois field multiplication over a different Galois field which requires different sets of look-up tables. A recent improvement in Galois field multiplier systems or engines provides faster operation and reduced storage requirements but still faster, lower power and smaller designs are demanded.